reserve n,k for Nat,
  X for non empty set,
  S for SigmaField of X;

theorem Th10:
  for f be Functional_Sequence of X,ExtREAL, x be Element of X st
  x in dom(f.0) holds (superior_realsequence f)#x = superior_realsequence(f#x)
proof
  let f be Functional_Sequence of X,ExtREAL, x be Element of X;
  set F = superior_realsequence f;
  assume
A1: x in dom(f.0);
  now
    let n be Element of NAT;
A2: (F#x).n = (F.n).x by MESFUNC5:def 13;
    dom(F.n) = dom(f.0) by Def6;
    hence (F#x).n =(superior_realsequence(f#x)).n by A1,A2,Def6;
  end;
  hence thesis by FUNCT_2:63;
end;
